19 January 1995

Physics Letters B 343 (1995) 329-332

PHYSICS LETTERS B

Perturbative strong interaction corrections to the heavy quark semileptonic decay rate

Michael Luke a, Martin J. Savage b, Mark B. Wise’ a Department of Physics, University of Toronto, Toronto, Ontario, Canada MSS IA7

b Department of Physics, Carnegie Mellon University, Pittsburgh PA 15213, USA ’ Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA

Received 18 September 1994; revised manuscript received 9 October 1994 Editor: H. Georgi

Abstract

We calculate the part of the order aa correction to the semileptonic heavy quark decay rate proportional to the number of light quark flavors, and use our result to set the scale for evaluating the strong coupling in the order as term according to the scheme of Brodsky, Lepage and Mackenzie. Expressing the decay rate in terms of the heavy quark pole mass mQ, we find the scale for the MS strong coupling to be 0.07 mQ. If the decay rate is expressed in terms of the MS heavy quark mass mQ (ma) then the scale is 0.12 mQ. we use these results along with the existing calculations for hadronic 7 decay to calculate the BLM scale for the nonleptonic decay width and the semileptonic branching ratio. The implications for the value of IV,1

extracted from the inclusive semileptonic B meson decay rate are discussed.

Inclusive semileptonic B decay has received consid- erable attention both theoretically and experimentally.

In the limit where the b quark mass is much larger

than the QCD scale the B meson decay rate is equal to the b quark decay rate [ 11. Corrections to this first

arise at order (hQcD/mb)* and these nonperturbative corrections may be written in terms of the matrix ele-

ments [2-41 (B16(iD)2blB) and (B16iga,,G~YblB). The measured semileptonic B decay rate provides a method for determining the magnitude of the element

of the Cabibbo-Kobayashi-Maskawa matrix I&. To one loop, the b quark decay rate is

r(b -+ x&e,) = I&j* 2 f(m/mb>

x [l--2 (d-T+&(mc/mb)) +...I. (1)

In equation ( 1) mb and m, are the pole masses of the b and c quarks, f(x) is defined by

f(x) = ( 1 - x4) ( 1 - Xx* + x4) - 24x4 In x (2)

and St takes into account the effects of the charm

quark mass on the order crs contribution to the b quark

decay rate [ 61. For m,/mb = 0.3, Sr = -1.11. In Eq. (1) the scale of the strong coupling as is

usually taken to be N mb. The size of the order cr$ cor- rection depends critically on this choice. If all of the

higher order terms in the (Y, expansion were known

then the decay rate would be independent of the choice of scale. However, some choices of scale give pertur- bation series that are badly behaved with higher orders in the coupling being very important. Brodsky, Lepage and Mackenzie (BLM) [ 51 have advocated choosing the scale so that vacuum polarization effects are ab-

0370-2693/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved. SSDIO370-2693(94)01370-5

330 hf. Luke et al. /Physics Letters B 343 (1995) 329-332

sorbed into the running coupling. This physically ap- pealing choice of scale usually results in a reasonable

perturbation series. In this letter we use our calcula-

tion of the part of the LY: correction proportional to nf

to determine the BLM scale appropriate for semilep-

tonic heavy quark decay.

Smith and Voloshin [7] have recently shown that the nf dependent part of the order LYE contribution to

the semileptonic decay rate for a heavy quark may be

written in terms of the one loop corrections evaluated with a finite gluon mass:

gr(2) = f n a$“)(mg) 677

00

X s( P(p) - mt? 0

b2 + mi> r”)(o) @

> P 2 ’

(3)

where I’(‘) (1~) is the order LY, contribution to the decay rate computed with a gluon of mass p, and

CZ(“) (me) is the strong coupling evaluated in the V- siheme of Brodsky, Lepage and Mackenzie. This is

related to the usual MS coupling &(mb) by [ 51

a(“)(&4 = k(p) + zj 4~ s 5C@ (11 -$f> +....

(4)

An expression analogous to Eq. (3) also holds for the differential rate dI/dt, where t = (pe + P~)~. We have calculated 51t2) for semileptonic Q + X,eG,

decay with a massless quark q in the final state and mQ < mw. The contribution of the graphs containing

a virtual gluon loop to the differential rate dr/dt with

a massive gluon was calculated analytically while the integral over the c quark energy in the bremmstrahlung graphs was performed numerically. The infrared diver- gences were shown explicitly to cancel in the sum, and the final integral over the gluon mass was performed numerically. Finally, the t integral was also performed numerically to obtain the correction to the total rate. In the t -+ 0 limit we reproduce the results given in [7] for top quark decay in the limit m, > mw.

We find for the total decay width

In terms of the MS coupling Eq. (5) becomes

UQ + Xpe>

= lvQ,12 g{ 1 -

+&(F,, [322]+...}. (6)

In Eq. (6) the term proportional to nf can be ab-

sorbed into the order Ly, term if the scale is changed

from mg to ,uBLM, where

-3 PBLM = mQ exp

(,rr2 - 25/4) 13.221)

r 0.07 mQ. (7)

The BLM scale for inclusive heavy quark decay is therefore significantly smaller than the naive estimate

of mQ. In Fig. 1 we plot the BLM scale for the dif-

ferential rate dr/dt as a function of the squared in-

variant mass t of the lepton pair. At t = 0 we find

the scale ,UBLM = 0.12 mQ, which coincides with the BLM scale found in [7] for top quark decay in the limit mw < m,. As would be expected on physical

grounds, /JBLM decreases as the invariant mass of the lepton pair increases.

The expression for the width found in Eq. (6) is

given in terms of the pole mass mQ of the heavy quark. The BLM scale ~BLM is different from that found in

Eq. (7) if the rate is expressed in terms of the running MS heavy quark mass evaluated at mQ. Using [ 81

mQ = fiQ(mQ) 4 h(mQ)

1 + Tjy

+(16.11- 1.04nf) (F)2+...} (8)

the semileptonic decay rate becomes

M. Luke et al. /Physics Letters B 343 (1995) 329-332 331

0 0.2 0.4 0.6 0.8 1

t/mQ2

Fig. 1. The BLM scale for the partial width dr/dt (in terms of the pole mass me) as a function of the lepton pair invariant mass squared t.

r(Q -+ K@,) = lV,,l’

X

+ $f (%)’ [-4.58, +. ..}. (9)

NOW the scale /_JBLM for which vacuum polarization effects are absorbed into the strong coupling is

-3 PBLM = mu exp

(7s - 65/4) [ -4.581)

= 0.12 mQ. (10)

It has been argued [7] that a low BLM scale, in-

dicating large two-loop corrections when a, (me) is used as an expansion parameter, would be expected when relating a “long-distance” quantity such as the heavy quark pole mass to the “short-distance” decay

rate. However, our results show that even if the “short-

distance” MS heavy quark mass is used, the BLM scale /LBLM for the order 5, correction to semilep- tonic heavy quark decay is still significantly less than

mQ. For b decay (neglecting the charm quark mass) the scale is about 500 MeV and for c decay rate it is

only about 150 MeV. These low scales suggest that QCD perturbation theory cannot be used for inclusive semileptonic D or A, decay and that an accurate ex- traction of 1 V& 1 from the inclusive semileptonic B de- cay rate [ 91 is not possible without including all terms of order ti: (me) (and perhaps even higher orders in

ii$) in the theoretical expression for the semileptonic

decay rate. We also note that the BLM scale for inclusive

semileptonic heavy quark decay is somewhat smaller

(relative to the heavy quark mass) than the analo-

gous scale for hadronic r decay. From the two-loop expression for the inclusive r width [ lo],

I( r --f Y, + hadrons) =3 1+7

(

&(m,)

I(7 + v,Pce-)

+(6.340-0.379nf) (+)*+...), (11)

the BLM scale for the one-loop expression is found

to be exp( -3 x 0.379)m, = 0.32 m7. Therefore, al-

though inclusive r and c decays involve comparable energy scales, perturbative QCD is likely to be at best

applicable only to the former. It is straightforward to extend our results to the case

of nonleptonic heavy quark decay to massless prod- ucts if we neglect the running of the Hamiltonian be- tween mw and p. At order Q,, the corrections to the

nonleptonic width are given by two classes of dia-

grams: those with gluons dressing the ?b vertex and

those with gluons dressing the & vertex. The first class is identical to that encountered in semileptonic decay, while the second gives the corrections to r de- cay. Combining Eqs. ( 11) and (6) and including the appropriate colour factors, we find the expression for the total nonleptonic width to massless quarks

332 M. Luke et al. /Physics Letters B 343 (1995) 329-332

+$f(=$)* [2.65]+...)

Because the one-loop correction to I’,] is much smaller

than for the semileptonic width Is], while the order

nfcuf terms are comparable, this results in a very low

BLM scale for the nonleptonic width:

-3 PBLM =mQexP (79 - 31/4)

P.651)

2( 0.02 mQ. (13)

Such a low scale should not be taken literally. It sim-

ply indicates that the two-loop corrections to r,t are significant, requiring an extremely low scale for cy, in

the one-loop term to absorb the order n& terms. We may also set the BLM scale for the semilep-

tonic branching fraction for decays to massless fermions [ 111. The contribution from the class of

graphs dressing the bc vertex cancels in the ratio

Isi / (I,,1 + I,,), and the corrections are given solely by the graphs which contribute to r decay ’ . We thus

find, independent of the our results for rst,

rsl 1 -=-

rnl+rsl 4

- 0.379nf (+q2+...]) (14)

(taking IV,,, I = 1) , which gives the same BLM scale relative to mQ as in r decay: /QLM = 0.32 mQ.

This work was supported in part by the U.S. Depart- ment of Energy under grants DE-FG03-92-ER40701

and DE-FGO2-91ER40682, and by the Natural Sci-

ences and Engineering Research Council of Canada.

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